Question

so sánh hai số bằng cách vận dụng hằng đẳng thức

A = 4(3²+1) (3⁴+1).....(3⁶⁴+1) và B = 3¹²⁸ -1

Answer 1

\(A=4\left(3^2+1\right)\left(3^4+1\right).....\left(3^{64}+1\right)\)

\(=\frac{1}{2}\left(3^2-1\right)\left(3^2+1\right)\left(3^4+1\right).....\left(3^{64}+1\right)\)

\(=\frac{1}{2}\left(3^4-1\right)\left(3^4+1\right).....\left(3^{64}+1\right)\)

\(=\frac{1}{2}\left(3^{128}-1\right)< B\)

Answer 2

\(A=4\left(3^2+1\right)\left(3^4+1\right)....\left(3^{64}+1\right)\)

\(\Rightarrow2A=\left(3^2-1\right)\left(3^2+1\right)\left(3^4+1\right).....\left(3^{64}+1\right)\)

\(=\left(3^4-1\right)\left(3^4+1\right).....\left(3^{64}+1\right)=\left(3^{64}-1\right)\left(3^{64}+1\right)=3^{128}-1=B\)

\(\Rightarrow A< B\)